ラゲール陪多項式（グラフ）

\(\normalsize Associated\ Laguerre\ polynomial\ L_n^\alpha(x)\\ (1)\ xy''+(\alpha+1-x)y'+ny=0\\ \hspace{25px} y=L_n^\alpha(x)\\ (2)\ {\large\frac{e^{-{\normalsize\frac{xt}{1-t}}}}{(1-t)^{\alpha+1}}}={\large\displaystyle \sum_{\small n=0}^{\small\infty}}L_n(x)t^n\\ (3)\hspace{5px}{\large\int_{\tiny 0}^{\tiny \infty}}L_n^\alpha(x)L_m^\alpha(x)e^{-x}x^\alpha dx={\large\frac{(n+\alpha)!}{n!}}\delta_{mn}\\ (4)\ L_n^\alpha(x)=(-1)^\alpha{\large\frac{d^\alpha}{dx^\alpha}}L_{n+\alpha}(x)\\ \hspace{65px}={\large\frac{e^xx^{-\alpha}}{n!}\frac{d^n}{dx^n}}(e^{-x}x^{n+\alpha})\\ (5)\ L_n^\alpha(x)={\large\frac{(\alpha+1)_n}{n!}}{}_{\small 1}F_{\small 1}(-n;\alpha+1;x)\\\)